The background description provided here is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventor, to the extent it is described in this background section, as well as aspects of the description that may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted as prior art against the present disclosure.
Traditional finite element method (or, traditional finite element analysis) was developed in the 1960s and is currently the best numerical method for evaluating continua and structures. It is usually used to address problems too complicated to be addressed with classical analytical methods.
One usage of finite element analysis is to evaluate stresses (internal forces in a body resulting from externally-applied loads) in structural components. Consider the plate in FIG. 1. An engineer could be presented with or have developed a design where the plate in FIG. 1 is welded in place around its center hole and has to carry the pressure and edge loading shown in FIG. 2 and FIG. 3. Note that the loads in FIGS. 2-3 are put on the plate simultaneously but are shown in different plots for clarity. The engineer might have also selected (or received a specification for) the metal to be used in the plate so that the material properties are known for the plate.
To be a well-engineered component, the engineer wants to make the plate thick enough to carry the loads without having it be permanently deformed or break. Due to factors such as added cost and added weight, the engineer also doesn't want to make the plate thicker than necessary. Considering the material properties, the engineer may decide that the plate can carry a certain stress (for example only, 36,000 psi) before the plate is in danger of permanent deformation or breaking.
Establishing the stress in the plate is well suited for finite element analysis. Finite element analysis establishes the stresses and strains in the plate by breaking the plate into many pieces, or elements. The collection of elements is called a mesh. The elements are of a size and shape that can be numerically evaluated. The loads and the circular inner edge that is fixed in place are referred to as boundary conditions and are applied to the mesh. By simultaneously evaluating all of the elements at once, the stresses and strains can be approximated for the whole plate. In general, the finer the mesh is, the more accurate the stress results are.
FIGS. 4A-4B show an example of traditional finite element analysis results for this problem. In this case, shell elements are used. In general, shell elements are planar and the plate thickness is included in the element formulation (rather than in the physical element shape). In traditional finite element analysis, a shell element consists of a shape (like a triangle or quadrilateral) and points (called nodes) at corners and sometimes along an edge. The triangular elements in FIGS. 4A-4B have a node at each corner and at the center of each edge.
A node can translate or rotate and neighboring elements can share a node. Boundary conditions are applied to nodes and the elements numerically evaluate the stresses and strains that result from the node movement. On the edge that is fixed in place, the nodes are not allowed to move. In the rest of the model, the nodes move according to the stiffness of the elements relative to the applied loads. The nodal movements are the traditional finite element method's degrees of freedom. If a node can translate in the x-direction, y-direction, and z-direction and it can rotate about those same directions, it is said to have six degrees of freedom. The number of degrees of freedom in a finite element model determines how much computer computation is required to solve the problem.
FIG. 4A is a stress plot looking straight down on the plate. It is given in von Mises stress which is used for comparison with the 36,000 psi value defined earlier. The highest stresses are present at the central opening of the mounting plate and the lowest stresses are present at the outside corners. FIG. 4B is a displacement plot, where the deformation in the most positive z-direction is present at the central opening and the deformation in the most negative z-direction at the outside corners. The plate in this plot is rotated and the z displacement is magnified 75× to make it easier to see how the plate is deforming under the loading.
Considering the requirement that the stress be less than 36,000 psi, the engineer could ascertain that the plate in this example should be strengthened because it is overstressed (with a maximum stress of 4.361 e+04 psi, or 43,610 psi). For reference, the maximum displacement in the plate is 4.593e−03 inches or 0.004593 inches.
Element edges in the traditional finite element method must be a straight line between nodes. Consequently, many elements must be meshed (as in this example) to accurately approximate the curvature of a curved edge of the shape. The mesh in FIG. 4A is sufficient to produce accurate results. However, this comes with the cost of over 10,000 degrees of freedom that must be evaluated.
In FIG. 5A, a coarser mesh is applied, which makes the model more efficient to run—i.e., requiring less processing and memory resources. The gain in efficiency may be significant because the number of degrees of freedom to be evaluated is approximately 1/10 of the degrees of freedom of the fine mesh. However, the results are less accurate. Note in FIG. 5A that the circular edges are being followed less accurately (resulting in a jagged edge) and the stress results have significant inaccuracies. This inaccuracy is partially due to the elements not following the circular edge very well. It is also partially due to the elements' size and shape and how the numerical solution is formulated. If the mesh shown in FIG. 5A were the only mesh used to evaluate the problem, it would incorrectly appear that the stresses were acceptable (the maximum stress being approximately 26,840 psi). FIG. 5B similar shows a displacement plot generated using the coarse mesh.